In a right-angled triangle ABC with a right angle C, the outer angle at the apex A is 120 degrees

In a right-angled triangle ABC with a right angle C, the outer angle at the apex A is 120 degrees, AC + AB = 18 cm. Find AC and AB.

Given: right-angled triangle ABC, angle C = 90⁰, external angle at apex A = 120⁰, AC + AB = 18 cm.
Find: АС -?
AB -?
Decision:
Since the sum of the outer and inner angles at one vertex is 180⁰, then the angle A is 180⁰ – 120⁰ = 60⁰.
The sum of the angles of the triangle is 180⁰, which means angle B = 180⁰ – angle C – angle A = 180⁰ – 90⁰ – 60⁰ = 30⁰.
The leg of a right-angled triangle, opposite an angle of 30⁰, is equal to half of the hypotenuse. So AC = ½ * AB.
Substitute this equality into the well-known equality AC + AB = 18
½ * AB + AB = 18
3/2 * AB = 18, AB = 18 * 2/3 = 12 cm.
AC = ½ * AB = 12/2 = 6 cm.
Answer: AC = 6 cm, AB = 12 cm.